Estimates in $L_{p}$ for solutions of SPDEs with coefficients in Morrey classes

Morrey-Campanato estimates for the moments of stochastic integral operators and its application to SPDEs

In this paper, we are concerned with the estimates for the moments of stochastic convolution integrals. We first deal with the stochastic singular integral operators and we aim to derive the Morrey-Campanato estimates for the $p$-moments (for $p\ge1$). Then, by utilising the embedding theory between the Campanato space and Hölder space, we establish the norm of $C^{θ,θ/2}(\bar D)$, where $θ\ge0, \bar D=\bar G\times[0,T]$ for arbitrarily fixed $T\in(0,\infty)$ and $G\subset\mathbb{R}^d$. As an application, we consider the following stochastic (fractional) heat equations with additive noises \be…

On strong solutions of Itô's equations with $Dσ$ and $b $ in Morrey classes containing $L_{d}$

We consider Itô uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in $W^{1}_{2+\varepsilon,loc}$, and the drift in a Morrey class containing $L_{d}$. We prove the unique strong solvability in the class of admissible solutions for any starting point. The result is new even if the diffusion is constant.

The parametrix method for parabolic SPDEs

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Ito-Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.

Scaling Limits of Solutions of SPDE Driven by Lévy White Noises

Consider a random process s solution of the stochastic partial differential equation Ls = w with L a homogeneous operator and w a multidimensional Lévy white noise. In this paper, we study the asymptotic effect of zooming in or zooming out of the process s. More precisely, we give sufficient conditions on L and w so that the rescaled versions of s converges in law to a self-similar process of order H at coarse scales and at fine scales. The parameter H depends on the homogeneity order of the operator L and the Blumenthal-Getoor indices associated to the Lévy white noise w. Finally, we apply ou…

Moment bounds in spde's with application to the stochastic wave equation

We exhibit a class of properties of an spde that guarantees existence, uniqueness and bounds on moments of the solution. These moment bounds are expressed in terms of quantities related to the associated deterministic homogeneous p.d.e. With these, we can, for instance, obtain solutions to the stochastic heat equation on the real line for initial data that falls in a certain class of Schwartz distributions, but our main focus is the stochastic wave equation on the real line with irregular initial data. We give bounds on higher moments, and for the hyperbolic Anderson model, explicit formulas f…

针对SPDEs局部波动的综合误差估计

Combined Error Estimates for Local Fluctuations of SPDEs

本文研究了某些抛物线随机偏微分方程类的局部波动的数值近似，重点分析小空间相关噪声在大偏差效应发生前的时间尺度上的影响，引入了一种新策略，证明了针对五个主要误差源的综合误差估计（CERES）：空间离散误差、局部线性化误差、噪声截断误差、局部弛豫误差到稳态以及通过迭代低秩矩阵算法的近似误差。